I Development of the new land surface scheme SiBUC 3

Transcription

1 Contents I Development of the new land surface scheme SiBUC 3 1 Basic concept of the SiBUC model The energy budget at different surface condition Heterogeneity of land surface and mosaic approach Surface elements and fractional area Urban Canopy model Street canyon geometry (k-story building) Radiation transfer process partition of incident radiation Radiation budget at roof surface Radiation budget at wall surface Radiation budget at road surface Upward radiation from street canyon Total net radiation and upward radiation Turbulent transfer process Wind profile and momentum flux Aerodynamic resistance Sensible heat flux Rainfall interception and interception loss Prognostic variables and their governing equations Governing equations for temperatures Governing equations for intercepted water Numerical solution of prognostic equations Water Body model Energy balance at water surface Albedo and radiation budget Turbulent transfer and surface fluxes Prognostic variables and their governing equations Numerical solution of prognostic equations Green area model Brief description of SiB model Model philosophy Structure of the SiB Atmospheric boundary conditions for SiB Prognostic physical-state variables of SiB Structure of the green area model

2 2 CONTENTS 4.3 Prognostic equations of the green area model Governing equations for temperatures Governing equations for intercepted water Governing equations for soil moisture stores Radiative transfer (two-stream approximation) Turbulent transfer and aerodynamic resistances Above the transition layer (z t z z m ) Within the transition layer (z 2 z z t ) Within the canopy air space (CAS) (z 1 z z 2 ) Below the canopy (z s z z 1 ) Solution of momentum transfer equation set Aerodynamic resistances Surface resistances Sensible and latent heat fluxes Numerical solution of prognostic equations A List of simbols and their definitions 57

3 Part I Development of the new land surface scheme SiBUC 3

4

5 Chapter 1 Basic concept of the SiBUC model 1.1 The energy budget at different surface condition The radiative energy absorbed from the sun and the atmosphere is partitioned into fluxes of sensible, latent, and gound heat, and this partitioning (redistribution of absorbed energy) is strongly dependent on both the land cover characteristics and its hydrological state. When the surface is wet, as is common in irrigated agricultural areas and after rain events, the incoming radiation is mostly used for evapotranspiration. In that case, sensible heat flux and soil heat flux are usually much smaller than latent heat flux, and Bowen ratio is close to zero. On a bare dry land, the absorption of the incoming radiation results in a strong heating of the surface, which usually generates a strong turbulent sensible heat flux and large soil heat flux. In that case, there is no evaporation, and the Bowen ratio is infinite. When the surface is covered by a dense vegetation, water is extracted mostly from the plant root zone by transpiration. Thus, latent heat flux is dominant even if the soil surface is dry, but as long as there is enough water available in the root zone and plants are not under stress conditions. In the case of water body, as will be seen in Chapter 3, significant part of absorbed energy is not used in a diurnal cycle of energy budget, and handed over in a seasonal cycle. The diurnal variation of energy budget and surface temperature tend to be small. Therefore, the energy budget components of water body have seasonal lag compared to those of other surfaces. In the case of urban area, the surface is covered by inpermeable pavement which does not allow the evaporation of soil moisture. Then, strong heating of the surface occurs as is the case for bare soil. Due to the complex geometrical structure of urban canopy, as will be seen in Chapter 2, the characteristics of turbulent transfer process and the radiation absorption and exchange process are much different from other surfaces. Furthermore, artificial heat is released as an additional energy as a result of human activity. The urban thermal environment depends not only on the difference in physical properties such as reflectance and heat capacity but also on the geometrical structure. Owing to the multiple scattering effects within street canyon (between street and wall, between facing walls), the albedo of of urban area becomes smaller than flat surface (Aida, 1978). As street canyon becomes deeper, skyview factor from wall and road becomes small, and propotion of long-wave radiative energy trapped inside the canyon becomes larger. 5

6 6 CHAPTER 1. BASIC CONCEPT OF THE SIBUC MODEL 1.2 Heterogeneity of land surface and mosaic approach At the grid resolution of atmospheric models, land-surfaces are very heterogeneous. This can be easily understood by looking at maps of soil, vegetation, topography, land-use patterns, etc. Figure 1.1 shows a land-use map of Lake Biwa Basin, Japan. The domain of this area is 1 degree 1 degree. Water surface and urbanized area are shown in white and blue color respectively. Figure 1.1: Land-use of Lake Biwa Basin (from KS-202) In the numerical weather prediction model, the size of grid element is usually decided from the need of high accuracy and high computational efficiency. If the grid size is selected to eliminate the heterogeneity of land-surface, in another word, if the grid size is selected according to the minimum size of land-use patch, it will become extremely small, and the number of grid elements will become enormous. Therefore, the problem of heterogeneity, off course, must be treated in

7 1.3. SURFACE ELEMENTS AND FRACTIONAL AREA 7 LSS (Land-Surface Scheme). One solution is to find effective parameters to generate representative values of the whole grid square. A great deal of work has already been done on aggregation research (Michaud and Shuttleworth, 1997). Sometimes such a parameter aggregation is not possible because of high non-linearity between parameter and resulting fluxes. Another way is to apply a mosaic type parameterization (Avissar and Pielke, 1989 ; Kimura, 1989). Such approach couples independently each land-use patch of the grid element to the atmosphere. The grid averaged surface fluxes are obtained by averaging the surface fluxes over each land-use weighted by its fractional area. 1.3 Surface elements and fractional area The SiBUC (Simple Biosphere including Urban Canopy) model presented here uses mosaic approach to incorporate all kind of land-use into LSS. In the SiBUC model, the surface of each grid area is divided into three landuse categories and five components. 1. green area (vegetation canopy(c), ground(g)) 2. urban area (urban canopy(uc), urban ground(ug)) 3. water body (wb) Boundary Condition : T m e mu mf Λ,µ (0) F τ, d(0) P Zm (Reference Height) Mc Tr Mr Tw Tc Canopy Urban Canopy Tdu Mw Mug Urban Ground Tug Twb Tdw Water Body Mg Tg Tdg W W 1 2 Ground Surface Layer Root Zone W 3 Recharge Zone Figure 1.2: Schematic image of surface elements in SiBUC. The schematic image of surface elements is shown in Figure green area is a natural land surface area (forest, grassland, bare soil, etc.), and it is usually treated in most LSS. 2. urban area is a artificial unpermeable surface (buildings, houses, pavement, etc.). 3. water body is a inland water surface (ponds, rivers, lakes) or ocean surface.

8 8 CHAPTER 1. BASIC CONCEPT OF THE SIBUC MODEL Although the surface of real basin is a mixture of much more constituents, all surface elements must be classified into either of them. All the surface elements that are included in the same category are lumped and treated as unit tile. Each gridbox has specified fixed fractions of these three land-use (V ga,v ua,v wb, respectively). Also canopy fractions within green area and urban area are also specified (V c,v uc, respectively). V ga + V ua + V wb =1, V ga 0, V ua 0, V wb 0 (1.1) 0 V c 1, 0 V uc 1 (1.2) Sensible heat, latent heat, and momentum fluxes are calculated separately by each sub-model (green area, urban canopy, water body). And the grid averaged surface fluxes are obtained by averaging the surface fluxes over each land-use weighted by its fractional area. [F ] total = i [F ] i V i =[F ] ga V ga +[F ] ua V ua +[F ] wb V wb (1.3) green area For the vegetation scheme, SiB (Sellers et al.,1986 ) is adopted. Some modification (simplification) from original SiB was done. The basic component of green area model will be described in Chapter 4. Now SiB2 (Sellers et al., 1994) can also be used as green area model (according to user s option). urban area A new urban canopy scheme which can account for complex structure of canyon geometry will be developed in Chapter 2. In the urban area, each roughness element is expressed by a square prism. All prisms are assumed to have the same width and to be evenly spaced, while they have their own roof heights. This description of urban canopy is used in the models for turbulent transfer. Radiation process is described precisely based on the sky-view factor. water body Water body scheme is developed in Chapter 3 based on force-restore model. Both diurnal and seasonal cycles of temperature are reproduced by this model. The concept of SiBUC was firstly proposed by Tanaka et al.(1994). At that time, the definition of each fractional area are slightly differnt from current version. Also, sub-model for urban canopy was too simple in the original version. In this study, urban-canopy model is entirely replaced based on recent works. The atmospheric boundary conditions (forcing variables) for SiBUC are the same as those for SiB and other land-surface schemes. And SiBUC has prognostic physical-state variables for each sub-model. Those are five surface temperatures, three deep layer temperatures, five interception water stores, and three soil moisture stores (see Table 1.1). When the grid size is small (less than one kilometer), the arrangement and size of each land-use within grid area have small effects on heat fluxes, and only the fractional area must be considered to express the land-use heterogeneity. But when the grid size is large (more than ten kilometers), the land-use scale effects may appear, and the arrangement and size of each land-use should be considered too. This scaling problem will be discussed in Part II. For the moment, SiBUC model just uses mosaic parameterization to calculate the grid average fluxes, and its application is limited to basin-scale. SiBUC can be used for the understanding of basin-scale energy and water cycles. SiBUC will be used as a tool for investigating the scaling problem.

9 1.3. SURFACE ELEMENTS AND FRACTIONAL AREA 9 Table 1.1: Prognostic variables and boundary conditions for SiBUC Prognostic Variables green area T c temperature for vegetation canopy K T g temperature for soil surface K T d temperature for deep soil in green area (daily mean of T g ) K M c interception water stored on canopy m M g interception water puddled on the ground m W 1 soil moisture wetness for surface layer W 2 soil moisture wetness for root zone W 3 soil moisture wetness for recharge layer urban area T r temperature for building roof K T w temperature for building wall K T ug temperature for road K T du temperature for deep soil in urban area (daily mean of T ug ) K M r interception water stored on building roof m M w interception water stored on building wall m M ug interception water puddled on the road m water body T wb temperature for water surface K T dw temperature for deep water (daily mean of T wb ) K Boundary Conditions z m reference height m T m air temperature at z m K e m vapor pressure at z m mb u m wind speed at z m ms 1 S downward short-wave radiation Wm 2 L downward long-wave radiation Wm 2 P precipitation rate ms 1

10

11 Chapter 2 Urban Canopy model In this chapter, urban canopy model to be used as sub-model of land-surface scheme in mesoscale atmospheric model is developed. Due to the complexity and diverity of urban area, the model is formulated to be as general as possible to represent any kind of city. Masson (2000) has also presented a physically-based urban canopy scheme based on canyon geometry. The model presented here is different in the treatment of canyon structure. The model allows the existence of building elements of different height. Radiation and turbulent transfer process in the urban area, where various depth of street canyons are mixed, is modelled and formulated numerically. Then, the effects of geometrical structure of city elements on energy and radiation environment in urban area are discussed. 2.1 Street canyon geometry (k-story building) F0b θ Vuc : 1-Vuc roof skyg skyw bw wall kho road Figure 2.1: schematic image of street canyon, direct beam radiation, and sky-view factor Figure 2.2: arrangement of street canyon (overlook from the sky) The canyon concept, developed by urban climatologists (e.g., Oke(1987)), uses such a framework: it considers a single road, bordered by facing buildings. Figure 2.1 shows the schematic image of street canyon, which is a basic component in the urban canopy model. In this figure, street canyon consists of building (width is b w, height is kh 0 (k-story)) and road. If we consider the actual situation of city, road orientation (and also orientation of building wall) is mixed. 11

12 12 CHAPTER 2. URBAN CANOPY MODEL If we have detailed information about individual orientation and location of each buildings and roads, it may be possible to calculate energy and radiation budget for all individual elements. But in usual case, these information are difficult to obtain. Even though the street pattern is relatively regular, the directions of solar radiation and wind are changing in time. Then, we abandon to describe individual orientations of each street canyons. In the calculation of radiation budget, it is assume that street is always perpendicular to the solar beam radiation (see Figure 2.1), and that buildings are located along identical roads, the length of which is considered far greater than their width (see Figure 2.2). Table 2.1: Variables and parameters in urban canopy model symbol definition unit V ua fraction of urban area in a grid area V uc building coverage in V ua n highest number of building story k number of building story in a canyon r(k) fraction of k-story building h 0 unit height of story m b w building width m skyw, skyg sky-view factor for wall and road rad C d drag coefficient α i (i = r, w, ug) reflectance (albedo) ε i (i = r, w, ug) emissivity c i (i = r, w, ug) specific heat Jm 3 K 1 k i (i = r, w, ug) thermal conductivity Wm 1 K 1 θ incident angle of direct beam radiation rad ds i,us i (i = r, lw, rw, ug) downward and upward short-wave radiation Wm 2 dl i,ul i (i = r, w, ug) downward and upward long-wave radiation Wm 2 Rn i (i = r, w, ug) net radiation Wm 2 H i (i=r,w,ug) sensible heat flux from roof, wall, road Wm 2 E i (i=r,w,ug) rates of evaporation from roof, wall, road ms 1 P i (i=r,w,ug) rate of rainfall interception by the roof,wall, road ms 1 D i (i=r,w,ug) rate of drainage from the roof, wall, road ms 1 F 0bi (i = r, w, ug) direct beam component of short-wave radiation Wm 2 F 0di (i = r, w, ug) diffuse component of short-wave radiation Wm 2 F 0ti (i = r, w, ug) long-wave radiation (diffuse only) Wm 2 P rainfall rate ms 1 u m wind speed at reference height K T m temperature at reference height K e m vapor pressure at reference height hpa T i (i = r, w, ug, du, bi) temperature K e (T i ) (i=r,w,ug) saturation vapor pressure at T i hpa M i (i=r,w,ug) water held on the roof, wall, road m S i (i=r,w,ug) maximum value of M i m W i (i=r,w,ug) wetted fraction of roof, wall, road r i (i = r, w, du, au) aerodynamic resistance sm 1 where subscript ( r, w, lw, rw, ug, du, bi) stands for roof, wall, left-side wall, right-side wall, road, soil, inside of building, respectively.

13 2.2. RADIATION TRANSFER PROCESS 13 Basically, it is assumed that radiation energy is emitted from the center point of each component (roof, wall, road). And also sky-view factor is defined as the view angle from these center points. Using building coverage (or plan area density) (V uc ), the width of canyon air space is expressed as b w (1 V uc )/V uc. Therefore, sky-view factor for wall (skyw) and road (skyg) in a k-story canyon is expressed as follows. kh 0 skyw = arccos ) 2 ( ) (2.1) 2 ( kh0 bw (1 V uc ) V uc kh 0 skyg = 2 arccos ( ) (2.2) 2 bw (1 V (kh 0 ) 2 uc ) + 2V uc Direct beam component of short-wave radiation (F 0b ) incidents to the canyon with zenith angle θ, diffuse component of short-wave and long-wave radiation insident hemispherically. 2.2 Radiation transfer process In this model, three components (roof, wall, road) are considered in the calculation of net radiation. Although there are various height of street canyon in the urban area, all canyons which have same height (k-story) are treated together. Assuming the situation of k-story canyon, radiation budget is calculated. Then, total net radiative energy for roof, wall, and road are calculated by integrating with roof height distribution function (r(k)). This is a basic concept of radiation process in urban canopy model. Firstly, the partition of long-wave and short-wave radiation incident to each element is described partition of incident radiation x direct beam F0b y θ F0d, F0t roof roof left wall φ ψ right wall β ν wall road reflection of direct beam road reflection of diffuse radiation Figure 2.3: reflection to wall surface

14 14 CHAPTER 2. URBAN CANOPY MODEL Fraction of direct beam radiation incident to roof is same as building coverage (V uc ). As for wall and road, radiation is partitioned according to the incident zenith angle (θ). This partition is x y in Figure 2.3. where, x : y = kh 0 tan θ : b w(1 V uc ) kh 0 tan θ (2.3) V uc Furthermore, radiation energy is partitioned according to the situation that the beam radiation incident to road or not. kh 0 when cos θ ( ) 2 (kh0 ) 2 bw (1 V uc ) + V uc ( ) h0 F 0br = F 0b V uc F 0bw = F 0b V uc k tan θ F 0bg = F 0b (1 V uc ) F 0bw (2.4) b w kh 0 when cos θ ( ) 2 (kh0 ) 2 bw (1 V uc ) + V uc F 0br = F 0b V uc F 0bw = F 0b (1 V uc ) F 0bg = 0 (2.5) Although diffuse component of short-wave radiation and long-wave radiation incident to street canyon hemispherically, these radiations are partitioned according to the fraction viewed from high above the sky (see Figure 2.2). So no difuse radiation component is assigned to the wall. F 0dr = F 0d V uc F 0dw =0 F 0dg = F 0d (1 V uc ) (2.6) F 0tr = F 0t V uc F 0tw =0 F 0tg = F 0t (1 V uc ) (2.7) Radiation budget at roof surface In fact, roof may obtain some of radiation from neighboring buildings. But we don t use the detailed information about the arrangement of individual streets and buildings. Then, direct interaction of radiation between each street canyon is omitted in the model. For the roof surface, the formulation of radiation budget becomes very simple, because it is not necessary to consider the radiation which is reflected or emitted from other element. ds r = F 0br + F 0dr us r = α r (F 0br + F 0dr ) (2.8) dl r = F 0tr ul r = σɛ r Tr 4 V uc (2.9) Rn r = ds r + dl r us r ul r (2.10) Radiation budget at wall surface In case of wall surface, the radiation scattered from facing wall and road are considered (single scatter only). In this model, the direction of street canyon is not considered, and it is described two dimensionally (see Figure 2.1, 2.2). And the wall which receive the direct beam is treated as left wall (right wall is shadow). In fact, the wall and street face to all directions. When the sun moves, some of wall which faces to the sun will receive the direct beam, and others not. This direct beam receiving wall is called as left wall. The radiation incident to the left wall (ds lw ) consist of direct beam from sun and reflected radiation from road (reflection from right wall is omitted (double scatter)).

15 2.2. RADIATION TRANSFER PROCESS 15 In the calculation of road reflection, view angle from the center point of direct beam receiving zone and diffuse radiation receiving zone are used (see Figure 2.3). These angles (φ and β in Figure 2.3) are expressed as follows. φ = arctan 2kh 0, β = π skyg b w (1 V uc ) 2 kh 0 tan θ V uc (2.11) The radiation incident to the right wall (ds rw ) consist of reflected radiation from left wall and road. In the calculation of road and wall reflection, view angle from the center point of direct beam receiving zone and left wall are used (see Figure 2.3). These angles (ψ and ν in Figure 2.3) are expressed as follows. ψ = arctan 2kh 0, ν = π 2skyw (2.12) b w (1 V uc ) + kh 0 tan θ V uc Reflection at wall and road is expressed by reflectance (α w,α g ), and only single scattering is treated. π 2skyw ds lw = F 0bw + F 0dw + α w F 0dw + α g F 0bg ( φ π π )+α π skyg gf 0dg (2.13) 2π us lw = α w (F 0bw + F 0dw ) (2.14) ds rw = F 0dw + α w (F 0bw + F 0dw ) π 2skyw + α g F 0bg ( ψ π π )+α π skyg gf 0dg (2.15) 2π us rw = α w F odw (2.16) Note that F 0dw in above equations are assumed to be zero. left wall 1. direct beam left wall 2. direct beam road (φ) left wall 3. diffuse road (β) left wall right wall 1. direct beam left wall (ν) right wall 2. direct beam road (ψ) right wall 3. diffuse road (β) right wall In case of wall, no long-wave radiation is received directly from the sky. Long-wave radiation emitted from road and facing wall are received according to the view angle (β and ν). In this case, left and right wall are treated in the same way. As for emission from wall, ratio of wall area to road area is multiplied. dl w = σɛ ug Tug 4 π skyg (1 V uc )+σɛ w T 4 π 2skyw kh 0 w (1 V uc ) 2π π (1 V b uc) w V uc (2.17) ul w = σε w Tw 4 (1 V kh 0 uc) (1 V b uc) w V uc (2.18) Rn w = ds lw + ds rw +2dL w us lw us rw 2uL w (2.19) Radiation budget at road surface There are two cases for direct beam radiation to incident to the road or not, depending on the solar angle. When there is no direct beam on the road, F 0bg in eq.(2.20), (2.21) becomes zero.

16 16 CHAPTER 2. URBAN CANOPY MODEL As for the fraction of reflected radiation from the wall, view angle is from the center point of the wall. Note that the direct beam does not incident to left wall, F 0bw in eq. (2.20) is only for left wall. Reflected short-wave radiation and emitted long-wave radiation are described in the same way as mentioned above. ds ug = F 0bg + F 0dg + α w (F 0bw +2F 0dw ) skyw π (2.20) us ug = α ug (F 0bg + F 0dg ) (2.21) dl ug = F 0tg +2σɛ w Tw 4 skyw π (1 V kh 0 uc) (1 V b uc) w V uc (2.22) ul ug = σε ug Tug 4 (1 V uc) (2.23) Rn ug = ds ug + dl ug us ug ul ug (2.24) Upward radiation from street canyon Some of the incident radiation is reflected by reflectance (α). And, as mentioned above, some of the reflected radiation from wall and road will be received again by another element, and the rest of the reflected radiation will return to the sky. The partitioning of these radiations is described in and The total of these returned radiations is upward short-wave radiation from the street canyon to the sky (us m ). Ratio of us m to incident radiation (above the roof level) is the bulk albedo of street canyon system (roof-wall-road union). us m = us r +(us lw + us rw ) skyw π + α ug F 0dg skyg π + α ugf 0bg ( 1 φ + ψ π ) (2.25) All the emitted long-wave radiation from roof will return to the sky. Long-wave radiation emitted from wall and road will return to the sky according to the sky-view factor (skyw, skyg). The total of these returned radiations is upward long-wave radiation from the street canyon to the sky (ul m ). ul m = ul r +2uL w skyw π + ul ug skyg π (2.26) Total net radiation and upward radiation The radiation components presented above are those for the k-story street canyon. If we consider the various height of canyon, the total radiation absorbed by roof, wall, and road and total upward radiation from canyon system should be integrated using the roof height distribution (fractional area of each canyon). Rn r,total = us m,total = n Rn r,k r(k) n Rn w,total = Rn w,k r(k) n Rn ug,total = Rn ug,k r(k) (2.27) k=1 k=1 k=1 n us m,k r(k) n ul m,total = ul m,k r(k) (2.28) k=1 k=1 where n r(k) = 1 (2.29) k=1

17 2.3. TURBULENT TRANSFER PROCESS Turbulent transfer process In this section, aerodynamic resistences are formulated according to the assumed arrangement of roughness element (buildings) and turbulent transfer theory. bw kh0 bw x story n k+1 k n Σ r(k) = 1 k=1 a(k) 2 1 x r(1) r(2)... r(k) r(k+1)... r(n) Figure 2.4: Arrangement of roughness element Figure 2.5: Roof height distribution In the calculation of turbulent transfer process, each roughness element is expressed by a square prism (see Figure 2.4). All prisms are assumed to have the same width (b w ) and to be evenly spaced, while they have their own roof heights (k h 0 for k-story building). For simplicity, a(k) is defined as a total of roof height distribution (r(k)) from k-story to the top (n-story). And also A b is defined as a total of roof height distribution (r(k)) multiplied by the height (k). a(k) = A b = n r(i) (2.30) i=k n r(k) k (2.31) k=1 Where, r(k) is fraction of k-story building. n r(k) = 1 (2.32) k=1 If the horizontal scale of a grid area is x, total roof area of k-story buildings is V uc r(k)( x) 2. While roof area of one building is b 2 w. Then number of k-story building is V uc r(k)( x b w ) 2. Therefore, total building frontal area per unit area (S s ) and total building surface area per unit area (S r ) are expressed as follows. S s = n ( ) x 2 ( ) b w kh 0 V uc r(k) /( x) 2 h0 = V uc A b k=1 b w b w (2.33) S r = V uc +4S s (2.34) The building frontal area density in k-story layer per unit area (A s (k)) is n ( ) x 2 ( ) A s (k) = b w h 0 V uc r(k) /( x) 2 h0 = V uc a(k) (2.35) b i=k w b w

18 18 CHAPTER 2. URBAN CANOPY MODEL Wind profile and momentum flux As the density of roughness elements increases, so does the roughness (z 0 ) of the surface. But a point comes where adding new elements merely serves to reduce the effective drag of those already present due to mutual sheltering. As Shaw and Pereira (1982) note, at this point the increase in drag is offset by an increase in zero-plane displacement (d 0 ) (Grimmond et al. (1999)). The dependence of z 0 and d 0 on the size, shape, density, and distribution of surface elements has been studied using wind tunnels, analytical investigations, and field observation. In this model, we use a formulation of z 0 and d 0 following Macdonald et al. (1998). In their formula, both horizontal fraction of roughness element (plan area density) and vertical section (frontal area density) are considered. In our model, we replace frontal area density with total building frontal area per unit area (S s ) and roughness height with mean building height (z ave ). z ave = A b h 0 (2.36) [ d 0 = z ave V uc (V uc 1) ] (2.37) [ z 0 = (z ave d 0 ) exp 0.5 C ( d 1 d ) ] S κ 2 s (2.38) z ave Assuming that the wind speed profile within canopy air space follows the exponential function by an attenuation coefficient a w, wind speed at the k-story layer (u(k)) is expressed as follows. ( u(k) =u 2 exp [ a w 1 k )] (2.39) n Where u 2 is wind speed at the top of the canopy (z = z 2 = nh 0 ). According to Macdonald (2000), a w can be approximated by linear relationship with frontal area density (a w =9.6S s ). As a first guess, we assume a neutral condition for describing the log-linear profile of wind speed above the canopy. Then, a friction velocity (u ) and u 2 are obtained from u m. κu m u = ln ( ) (2.40) z m d 0 z 0 u 2 = u ( ) κ ln z2 d 0 (2.41) As for the profile of vertical eddy diffusivity (K m ), it is propotional to the height in log-linear region (above canopy). And K m is propotional to the local wind speed within canopy. z 0 K m (k) = κu (kh 0 d 0 ) (k n) (2.42) K m (k) = ηu(k) (k n) (2.43) η is obtained from the continuity of K m at canopy top height (k = n) Aerodynamic resistance η = κu (nh 0 d 0 ) u 2 (2.44) Now aerodynamic resistances (r r,r w,r du, and r au ) can be calculated using the profiles of u and K m. Transfer pathways of sensible heat in the urban canopy model are shown in Figure from the roof to the canyon air space (r r )

19 2.3. TURBULENT TRANSFER PROCESS from the wall to the canyon air space (r w ) 3. from the road to the canyon air space (r du ) 4. from the canyon air space to the reference level (r au ) In case of urban canopy, wall (frontal area) is important for momentum absorption, and roof (surface area) is so for scalar (heat and water vapor) transfer. In case of vegetation canopy, leaf is important for both momentum and scalar transfer. This is a fundamental difference between urban canopy and vegetation canopy. Thus, in calculating bulk aerodynamic resistances of roof and wall, surface area densities (A rr (k), A rw (k)) are used instead of frontal area density (A s (k)). 1 r r = 1 r w = n k=1 n k=1 A rr (k) U(k) h 0 = V uc p s C s p s C s A rw (k) U(k) h 0 = 4V uc p s C s p s C s n ( ) r(k) U(k) k=1 ( ) n h0 b w k=1 ( ) a(k) U(k) (2.45) (2.46) The extent to which the drag on individual roughness elements is reduced by the presence of neighbours was expressed by Thom (1971) in terms of a shelter factor. But shelter factor (p s )is still not well understood. It accounts for the observation that the drag coefficient of an ensemble of roughness elements is less than the sum of their individual drag coefficients, presumably due to mutual sheltering effects. Using the same relationship as the case for vegetation canopy, p s is expressed by power function, exept that leaf area density is replaced by frontal area density. p s =1+ ( Ss ) 0.6 ( ) Vuc A 0.6 b =1+ (2.47) nh 0 nb w This relationship should be investigated for the array of different height of roughness elements. The road to canopy air space resistance (r du ) is defined as follows. r du = ha 0 1 k ha h 0 dz = K m k=1 K m (k) (2.48) Canopy source height (h a = k ha h 0 ) is assumed to be equal to the center of action of r r and r w within the canopy as obtained from the solution of k ha k=1 ( r(k)+4 ( ) h0 b w a(k)) U(k) n k=k ha ( r(k)+4 ( ) h0 b w a(k)) U(k) (2.49) Resistance between canopy air space and reference height (r au ) can be described by integration of K m over the distance from h a to z m. r au = nh0 1 zm 1 dz + dz = h a K m nh 0 K m n h 0 k=k ha K m (k) + 1 ln κu ( ) zm d 0 nh 0 d 0 (2.50) Above formulation is based on the neutral condition. The nonneutral adjustment to aerodynamic resistances is executed according to Monin-Obukov similarity (see next section).

20 20 CHAPTER 2. URBAN CANOPY MODEL 2.4 Sensible heat flux If we assume no storage of heat at any of the junctions of the resistance network shown in Figure 2.6, we can write the area-averaged sensible heat fluxes from the roof, wall, and road as follows. H r = A(T r T au ), A = ρ a C p /r r (2.51) H w = B(T w T au ), B = ρ a C p /r w (2.52) H ug = C(T ug T au ), C = ρ a C p /r du (2.53) H r + H w + H ug = D(T au T m ), D = ρ a C p /r au (2.54) Canyon air space temperature (T au ) can be eliminated by asuuming that energy fluxes from roof, wall, and road are all transfered to atmospheric boundary layer, in another word, no energy is stored in canopy air space. T au = A T r + B T w + C T ug + D T m A + B + C + D (2.55) Tm em Tr Roof Tw Wall Hr r r r w Hua Hw r du Tug r au Tau Hug e (Tr) * Mr Roof Wall Er r r Eua r w Ew Mug r du r au e au Eug e (Tug) * Urban Ground Urban Ground Figure 2.6: Transfer pathways of sensible heat flux and aerodynamic resistances Figure 2.7: Transfer pathways of latent heat flux and aerodynamic resistances Now atmospheric stability can be calculated from the temperature difference between canyon air space and reference height (T au T m ). H = ρ a C p (T au T m )κu /Ψ H (2.56) L = ρ a C p T m u 3 /κgh (2.57) ζ = z m /L (2.58) In the urban area, sensible heat flux is usually dominant to latent heat flux. Therefore, only sensible heat flux is considered in eq.(2.57). If ζ is not close to zero, wind profile used in the previous section should be modified by integrated universal function.

21 2.5. RAINFALL INTERCEPTION AND INTERCEPTION LOSS 21 when ζ<0 (unstable) ( ) zm d 0 Ψ M = ln +ln (x )(x 0 +1) 2 z 0 (x 2 + 1)(x +1) + 2 2(tan 1 x tan 1 x 0 ) (2.59) ( ) ( ) zm d 0 y0 +1 Ψ H = ln +2ln (2.60) z 0 y +1 x = (1 16ζ) 1/4,x 0 =(1 16ζ 0 ) 1/4,y=(1 16ζ) 1/2,y 0 =(1 16ζ 0 ) 1/2 (2.61) when ζ 0 (stable) ( ) zm d 0 Ψ M = ln + 7 z 0 3 ( ) zm d 0 Ψ H = ln ln z 0 ln 1+3ζ +10ζ 3 1+3ζ 0 +10ζ /400ζ ζ 2 1+7/400ζ ζ 2 0 (2.62) (2.63) u = κu m /Ψ M (2.64) Now u from eq.(2.40) (neutral value) is replaced by u from eq.(2.64) (nonneutral value), and go back to eq.(2.41). The above procedure is repeated until convergence condition (in ζ) is obtained. Note that Ψ H in eq.(2.56) is equal to ln ( ) z m d 0 z 0 for the first iteration step. 2.5 Rainfall interception and interception loss The interception of rainfall and evaporation of intercepted water is modeled very simply. Basically, each components of urban canopy model (roof, wall, road) is assumed to be impermeable. And, different from the case for vegetation, roof is only one layer (interception area is much smaller than vegetation canopy). The maximum values for water store (S i ) are set to each story. If the water store (M i ) exceeds the maximum value, the drainage (D i ) occurs. P r = PV uc (2.65) { = 0 when Mr <S D r = r (2.66) = P r when M r = S r P w = D r (2.67) { = 0 when Mw <S D w = w (2.68) = D r when M w = S w P ug = P (1 V uc )+D w (2.69) { = 0 when Mug <S D ug = ug (2.70) = P ug when M ug = S ug P VucP Er Dr Mr Roof Wall Ew Mw Dw Eug Mug Urban Ground (1-Vuc)P Dug D ug is the surface runoff from urban area. It will flow into water body, or it will be used as input of runoff model. Figure 2.8: Rainfall interception model

22 22 CHAPTER 2. URBAN CANOPY MODEL The quantities M r,m w,m ug are used to determine the fractional wetted areas of the roof, wall, and road (W r,w w,w ug ). So the rates of evaporation from the wetted portions of the roof, wall, and road are expressed as follows. λe r = E[e (T r ) e au ], E = ρ ac p γ λe w = F [e (T w ) e au ], F = ρ ac p γ λe ug = G[e (T ug ) e au ], G = ρ ac p γ λe r + λe w + λe ug = H[e (e au ) e m ], H = ρ ac p γ W r r r (2.71) W w r w (2.72) W ug r du (2.73) W r r au (2.74) (2.75) e au can be eliminated by asuuming that water vapor fluxes from roof, wall, and road are all transfered to atmospheric boundary layer, in another word, no water vapor is stored in canopy air space. e au = E e (T r )+F e (T w )+G e (T ug )+H e m E + F + G + H (2.76) W r = W w = W ug = { = Mr /S r when e (T r ) >e au (2.77) = 1 when e (T r ) e au { = Mw /S w when e (T r ) >e au (2.78) = 1 when e (T r ) e au { = Mug /S ug when e (T ug ) >e au (2.79) = 1 when e (T ug ) e au (2.80) It is assumed in eq.(2.71), (2.72), (2.73) that wet and dry parts of the surface are at the same temperature. Basically, the maximum water storage is much smaller than vegetation canopy, and the duration time of water to present on the surface is very short, it is better to use the same temperature for wet and dry parts. 2.6 Prognostic variables and their governing equations Urban canopy model has seven prognostic physical-state variables: four temperatures (for roof, T r, for wall, T w, for road, T ug, and for under ground, T du ); three interception water stores (for roof, M r, for wall, M w, and for road M ug ). Where T du is defined as mean value of T ug over one day Governing equations for temperatures The governing equations for the four temperatures are based on force-restore method (Bhumralkar, 1975; Blackadar, 1976). The force-restore approximation relies on the analytical solution of the heat conduction equation under periodic forcing, which is used to parameterize the almost periodic

23 2.6. PROGNOSTIC VARIABLES AND THEIR GOVERNING EQUATIONS 23 daily ground heat flux. In this way, a very simple and efficient but reasonably accurate description of the temperature dynamics can be achieved. In this model, prognostic equation for surface temperature incorporates both surface energy flux (atmospheric forcing term) and soil heat flux (restoring term) to reproduce the diurnal variation of surface temperature realistically. In case of the urban canopy model, three surface temperatures (T r,t w,t ug ) are predicted. As for the restoring term for T r and T w, a inner building temperature (T bi ), which may be controlled or maintained by human activity, is used. Furthermore, artificial heat source (Q m 1 ), which is a result of human activity, is added to the forcing term. So, the prognostic equations for temperatures in urban canopy model are expressed as follows. T r C r t T w C w t T ug C ug t = Rn r H r λe r ωc r (T r T bi )+Q m V uc (2.81) = Rn w H w λe w ωc w (T w T bi )+Q m A w (2.82) = Rn ug H ug λe ug ωc ug (T ug T du )+Q m (1 V uc ) (2.83) T du C du = Rn ug H ug λe ug ωc du (T du T bi )+Q m (1 V uc ) (2.84) t The effective heat capacities (C r,c w,c ug,c du ) are defined theoretically using specific heat and thermal conductivity. C i = ( ) 1/2 ci k i (i=r,w,ug) (2.85) 2ω C du = 365C ug (2.86) where Q m = anthropogenic heat source (Wm 2 ) C i (i = r, w, ug, du) = effective heat capacity (J m 2 K 1 ) As for eq. (2.86), heat capacity is propotional to a square root of cycle. T du is defined as mean value of T ug over one day, and T du is expected to have seasonal cycle (one year). Effective heat capacity is impotant for reproducing the amplitude and phase of diurnal cycle. Basically, actual values of thermal properties (c i,k i ) are dependent on material (concrete, asphalt, tile, etc.). In practice, these parameters are adjusted to reproduce the diurnal cycle, since urban canopy consists of different kinds of materials Governing equations for intercepted water The governing equations for the three interception water stores are expressed in the same formula. M i t = P i D i E i ρ w (i = r, w, ug) (2.87) This is simple water budget equation for water store. In the calculation of water store (M i ), evaporation (E i ) is modified if eq.(2.87) produces a negative value of M i. 1 Usually, T bi is controlled by air conditioner. Then, Q m should be related to T bi or temperature difference with outside (T bi T m ). Furthermore, additional energy will be released as a result of domestic use. There is no good way to formulate T bi and Q m at present. Therefore, they are treated as parameter (prescribed, changable with time). The role of Q m will be discussed in the later section (see??).

24 24 CHAPTER 2. URBAN CANOPY MODEL 2.7 Numerical solution of prognostic equations In the numerical solution of the prognostic equations for temperatures (T r,t w,t ug ), we make use of the fact that the storage terms, involving C i (i = r, w, ug), are small relative to the energy fluxes Rn i,λe i, andh i (i = r, w, ug). These values make eq. (2.81), (2.82), (2.83) fast response equations so that changes in T r,t w,t ug, even over a time step as short as an hour, can have a significant feedback on the magnitude of the energy fluxes. The energy fluxes are explicit functions of atmospheric boundary conditions, prognostic variables, and aerodynamic resistances. Prognostic equations are solved by an implicit backward method using partial derivatives of each term. First, considering the energy fluxes in prognostic equations are functions of temperature, partial derivatives are calculated in subroutine partial. Then, prognostic equations are expressed in explicit backward-differencing form and a set of linear simultaneous equations regarding the changes in temperatures over a time step ( t) are obtained. Not only energy fluxes but also heat exchange terms have dependency on temperatures. Now prognostic equations can be written in discrete-time form. C r T r t C w T w t C ug T ug t C du T du t = Rn r H r λe r ωc r (T r T bi )+Q m V uc + Rn r T r ( H r T r + H r T w + H r T ug ) T r T r T w T ug ( λe r T r + λe r T w + λe r T ug ) ωc r T r (2.88) T r T w T ug = Rn w H w λe w ωc w (T w T bi )+Q m A w +( Rn w T w + Rn w T ug ) ( H w T w + H w T r + H w T ug ) T w T ug T w T r T ug ( λe w T w T w + λe w T r + λe w T ug ) ωc w T w (2.89) T r T ug = Rn ug H ug λe ug ωc ug (T ug T du )+Q m (1 V uc ) +( Rn ug T ug ( λe ug T ug T ug + Rn ug T w ) ( H ug T w T ug + H ug T ug T r T r + H ug T w T w ) T ug + λe ug T r + λe ug T w ) ωc ug ( T ug T du ) (2.90) T r T w = Rn ug H ug λe ug + ωc du (T bi T du )+Q m (1 V uc ) +( Rn ug T ug ( λe ug T ug T ug + Rn ug T w ) ( H ug T w T ug + H ug T ug T r T r + H ug T w T w ) T ug + λe ug T r + λe ug T w ) ωc du T du (2.91) T r T w If it is written in matrix form, KX = Y X = K 1 Y K 1,1 K 1,2 K 1,3 K 1,4 K K = 2,1 K 2,2 K 2,3 K 2,4 X = K 3,1 K 3,2 K 3,3 K 3,4 K 4,1 K 4,2 K 4,3 K 4,4 T r T w T ug T du

25 2.7. NUMERICAL SOLUTION OF PROGNOSTIC EQUATIONS 25 where K 1,1 = Cr t Rnr T r K 1,3 = Hr T ug K 2,1 = Hw T r K 2,3 = Rnw T ug K 3,1 = Hug T r K 3,3 = Cug t K 4,1 = Hug T r K 4,3 = Rnug T ug + Hr T r + λer T r + ωc r K 1,2 = Hr T w T ug K 1,4 =0 + λer + λew T r + Hw T ug + λeug T r Rnug T ug + λeug T r + Hug T ug + λew + Hug T ug T ug K 2,4 =0 + λeug T ug + λer T w K 2,2 = Cw t Rnw T w K 3,2 = Rnug T w + λeug T ug + ωc ug K 3,4 = ωc g K 4,2 = Rnug T w K 4,4 = C du t + Hw T w + Hug T w + Hug T w + ωc du + λew T w + λeug T w + λeug T w + ωc w Y = Rn r H r λe r ωc r (T r T bi )+Q m V uc Rn w H w λe w ωc w (T w T bi )+Q m A w Rn g H ug λe ug ωc ug (T ug T du )+Q m (1 V uc ) Rn g H ug λe ug + ωc du (T bi T du )+Q m (1 V uc ) Above equations can be solved in terms of temperature changes ( T r, T w, T ug, T du ). Each temperatures are updated to the value at time t 0 + t by adding temperature changes to the initial value at time t 0. Furthermore, energy fluxes are modified to show the values averaged over a time step (between time t 0 and time t 0 + t). Rn r = Rn r + 1 Rn r T r 2 T r (2.92) Rn w = Rn w ( Rn w T w + Rn w T ug ) T w T ug (2.93) Rn g = Rn g ( Rn ug T w + Rn ug T ug ) T w T ug (2.94) H r = H r ( H r T r + H r T w + H r T ug ) T r T w T ug (2.95) H w = H w ( H w T r T r + H w T w T w + H w T ug T ug ) (2.96) H g = H g ( H ug T r + H ug T w + H ug T ug ) T r T w T ug (2.97) λe r = λe r ( λe r T r + λe r T w + λe r T ug ) T r T w T ug (2.98) λe w = λe w ( λe w T r λe ug = λe ug ( λe ug T r T r + λe w T w + λe w T ug ) (2.99) T w T ug T r + λe ug T w + λe ug T ug ) (2.100) T w T ug

26 26 CHAPTER 2. URBAN CANOPY MODEL Following Section 2.2, five partial derivatives of net radiation fluxes (Rn r,rn w,rn ug ) are obtained as follows. Rn r = 4σε r Tr 3 T V uc (2.101) r Rn w = 4σε w T 3 π 2skyw n kh 0 n w (1 V uc ) T w π (1 V k=1 b uc) 4σε w T 3 kh 0 w(1 V uc ) (1 V w V uc k=1 b uc) (2.102) w V uc Rn w = 4σε g T 3 π skyg g (1 V uc ) (2.103) T g 2π Rn ug = 4σε ug Tug 3 T (1 V uc) (2.104) ug Rn ug = 8σε w T 3 skyw w T w π (1 V uc) n k=1 kh 0 b w (1 V uc) V uc (2.105) Following Section 2.4, nine partial derivatives of sensible heat fluxes (H r,h w,h ug ) are obtained as follows. T au = A T r + B T w + C T ug + D T m A + B + C + D H r A(B + C + D) = T r A + B + C + D, H r AB = T w A + B + C + D, H w B(A + C + D) = T w A + B + C + D, H w AB = T r A + B + C + D, H ug C(A + B + D) = T ug A + B + C + D, H ug AC = T r A + B + C + D, H r T ug = H w T ug = H ug T w = AC A + B + C + D (2.106) (2.107) BC A + B + C + D (2.108) BC A + B + C + D (2.109) Following Section 2.5, nine partial derivatives of latent heat fluxes (λe r,λe w,λe ug ) are obtained as follows. Where, e (T i) is a slope of saturation vapor pressure curve at T i. e au = E e (T r )+F e (T w )+G e (T ug )+H e m E + F + G + H λe r = e (T r)e(f + G + H) T r E + F + G + H, λe r = e (T w)ef T w E + F + G + H, λe w T w = e (T w )F (E + G + H) E + F + G + H, λe w T r = e (T r )EF E + F + G + H, λe ug T ug = e (T ug)g(e + F + H) E + F + G + H, λe ug T r = e (T r)eg E + F + G + H, (2.110) λe r T ug = e (T ug)eg E + F + G + H (2.111) λe w T ug = e (T ug )FG E + F + G + H (2.112) λe ug T w = e (T w)fg E + F + G + H (2.113)

27 Chapter 3 Water Body model In this chapter, water body model to be used as sub-model of land-surface scheme is formulated based on force-restore model. A dataset from flux measurement system of the Lake Biwa Project (see Appendix??) is used for the development and validation of the water body model. Performance of the produced model is also shown by numerical simulaton. Variables and parameters used in this chapter are listed in Table 3.1. Physical constants are listed in Table A.1 in Appendix A. Table 3.1: Variables and parameters in the water body model symbol definition unit z m reference height m u m wind speed at reference height ms 1 T m air temperature at reference height K e m vapor pressure at reference height mb q m specific humidity at reference height kg kg 1 µ cosine of zenith angle of incident beam T wb temperature of surface layer K T dw temperature of deep layer K q wb saturation specific humidity at T wb kg kg 1 Rn wb net radiation flux Wm 2 F s,wb short-wave radiation absorbed by water body Wm 2 H wb sensible heat flux Wm 2 E wb evaporation rate kg m 2 S 1 τ wb momentum flux absorbed by water surface kg m 1 s 2 u friction velocity ms 1 z 0 surface roughness m L Monin-Obukov length m ζ non-dimensional stability factor r aw aerodynamic resistance between water surface and reference height sm 1 C wb effective heat capacity of surface layer Jm 2 K 1 C dw effective heat capacity of deep layer Jm 2 K 1 Z s effective depth of surface layer m β short-wave radiation penetration factor α albedo of water surface α 1,α 2 parameter for albedo model (α = α 1 µ α 2 ) 27

28 28 CHAPTER 3. WATER BODY MODEL 3.1 Energy balance at water surface Figure 3.1 shows an example of seasonal variation of energy budget at the Lake Biwa in Japan. Horizontal axis is time (day) from 25th July in 1999 to 24th July in Upper figure is a time series of daily data, and lower one is a moving average of 15 days. The net radiation (Rn) has its maximum in July and minimum (nearly zero) in December. Very small net radiation in December can be explained by small sunshine and large upward long-wave radiation. Observation site is located at the north part of the Lake Biwa. Then, sunshine is small (almost cloudy) in winter season. Furthermore, due to the relatively warm temperature, upward long-wave radiation is larger than downward. In spite of much net radiative energy, latent heat (evaporation) is not so large from early spring to late summer. On the other hand, latent heat (evaporation) is large (even larger than net radiation) during autumn and winter season. What I want to say here is that there is a phase lag between seasonal cycle of net radiation (Rn), latent heat (λe), and sensible heat (H). This means that significant part of energy is not used (or released) in a diurnal cycle of energy budget, and handed over in a seasonal cycle. This kind of seasonal dynamics of energy budget is peculiar to water body, especially in deep lake. 300 Lake / /24 Heat Flux (W/m2) Rn(obs) le(stab) H(stab) G(res) Heat Flux (W/m2) Rn(obs) le(stab) H(stab) G(res) /9/ / /16 00/2/4 Time(day) 450 3/ / /3 Figure 3.1: Seasonal variation of energy budget at the north part of the Lake Biwa (1999/7/ /7/24). upper: daily average, lower: moving average of 15 days

29 3.2. ALBEDO AND RADIATION BUDGET Albedo and radiation budget Figure 3.2 shows the seasonal variation of lake surface albedo from 25th July in 1999 to 24th July in 2000 by moving average of 15 days. Albedo becomes large in winter season and small in summer season. This is deeply related to the change in the solar zenith angle. Figure 3.3 shows the relationship between the albedo (α) and cosine of zenith angle of incident solar radiation (µ). In this figure, data are selected according to the following criteria to see this dependency clearly. S 0.75 S top S 50 (Wm 2 ) µ 0.1 where S = downward short-wave radiation observed at the surface S top = downward short-wave radiation at the top of atmosphere (theoretical value) albedo Albedo of the Lake Biwa (1999/7/ /7/24) time (day) Albedo Albedo of the Lake Biwa y = x R = cosine of zenith angle of the incident beam Figure 3.2: Albedo of the Lake Biwa (moving average of 15 days, 1999/7/ /7/24) Figure 3.3: Dependency of albedo (α) on cosine of zenith angle of incident beam (µ) When the sky is overcast or partly clouded, the propotion of direct beam radiation component becomes smaller than that of diffuse component. In that case, albedo shows the weakly dependency on the solar zenith angle. Anyway, the reflectance of water surface has strong dependency on µ for direct beam component. In the SiBUC model, following Goudriaan (1971), downward short-wave radiation is divided into four components (F vb (visible-beam), F vd (visible-diffuse), F nb (nearinfrared-beam), and F nd (near-infrared-diffuse)) in the subroutine goudriaan (see Appendix??) by the need of vegetation model (SiB). Then, we can treat direct beam component and diffuse component separately. The radiation budget for the water surface can be written as follows. α b = aµ b (3.1) Rn wb = (F vb + F nb )(1 α b )+(F vd + F nd )(1 α d )+F td ε w σtwb 4 (3.2) Where, α b and α d are reflectance for direct beam component and diffuse component respectively. Unfortunately, direct beam component and diffuse component was not observed separately, we couldn t fix the relationship of eq.(3.1) directly. Actually, it is impossible to measure the propotion of reflected direct beam and reflected diffuse radiation. Therefore, the α b and α d are assumed to be equal, and the parameters in the relationship between reflectance and µ (eq.(3.3)) are decided through the the comparison of simulated and observed values (see Section??). α b = α d = α 1 µ α 2 (3.3)